Method for compressing digital images

ABSTRACT

The invention concerns a method for compressing data, in particular images, by transform, in which method this data is projected onto a base of localized orthogonal or biorthogonal functions, such as wavelets. To quantize each of the localized functions with a quantization step that enables an overall set rate R c  to be satisfied, the method includes the following steps:
         a probability density model of coefficients in the form of a generalized Gaussian is associated with each subband,   the parameters α and β of this density model are estimated, while minimizing the relative entropy, or Kullback-Leibler distance, between this model and the empirical distribution of coefficients of each subband, and   from this model, for each subband, an optimum quantization step is determined such that the rate allocated is distributed in the various subbands and such that the total distortion is minimal. As a preference, for each subband, the graphs of rate R and distortion D are deduced, from the parameters α and β, as a function of the quantization step and these graphs are tabulated to determine said optimum quantization step.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based on French Patent Application No. 02 05 724filed May 7, 2002, the disclosure of which is hereby incorporated byreference thereto in its entirety, and the priority of which is herebyclaimed under 35 U.S.C. §119.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method for optimal allocation of the bit rateof a data compressor by transform.

It concerns more specifically such an allocation process for acompressor by orthogonal or biorthogonal transformation, in particular awavelet transformation, used in combination with a scalar quantizer anda lossless entropy coder.

Hereinafter, reference will mainly be made to a wavelet transformationleading to a Multiresolution Analysis (MRA) implemented using digitalfilters intended to perform a decomposition into subbands. However, theinvention is not limited to a wavelet transformation.

It is recalled that an MRA consists in starting from an image in thespace domain with a set of image elements, or pixels, and in decomposingthis image into subbands in which the vertical, horizontal and diagonaldetails are represented. Thus there are three subbands per resolutionlevel as indicated in FIG. 1.

FIG. 1 illustrates an MRA on three resolution levels. In thisrepresentation, a subband is represented by a block. Thus, the image isfirst divided into four blocks with three subbands W_(1,1), W_(1,2) andW_(1,3) and a low-frequency representation W_(1,0) of the initial image.Subband W_(1,1) contains the horizontal wavelet coefficients; subbandW_(1,2) contains the vertical wavelet coefficients; subband W_(1,3)contains the diagonal wavelet coefficients and the block W_(1,0) iscalled “summary” or low frequencies.

At the next resolution level, the block W_(1,0) is itself divided intofour blocks (one summary and three subbands) W_(2,0), W_(2,1), W_(2,2)and W_(2,3) and, finally, the block W_(2,0) is divided into four blocksW_(3,0), W_(3,1), W_(3,2) and W_(3,3) for the third resolution level.Naturally, a finer division (by increasing the resolution levels) or acoarser division (by reducing the number of resolution levels) can becarried out.

2. Description of the Prior Art

It is known that the wavelet transform is well suited to imagecompression since it provides strong coefficients when the imageexhibits strong local variations in contrast, and weak coefficients inthe areas in which the contrast varies slightly or slowly.

It is also known that the probability distribution of a subband can bemodeled by a two-parameter unimodal function, centered at the origin, ofthe generalized Gaussian type:

${G_{\alpha\beta}(x)} = {\frac{\beta}{2\;{{\alpha\Gamma}\left( \frac{1}{\beta} \right)}}{\mathbb{e}}^{- {\frac{x}{\alpha}}^{\beta}}}$

where

Γ(ξ) = ∫₀^(+∞)𝕖^(−x)x^(ξ − 1) 𝕕x

For certain applications, particularly when the compression data must betransmitted over transmission channels imposing a bit rate, it isnecessary to quantize the subband coefficients in an optimal manner byminimizing the total distortion while satisfying a set bit rate.

The known optimum rate allocation methods propose, in general,performing a digital optimization process based on the minimization of afunctional linking rate and distortion and controlled by a Lagrangeparameter. In this case, an iterative optimization algorithm is employedwhich is generally very costly in calculation time and therefore cannotbe used for real-time applications or for applications with limitedcalculation resources. Only simplification of these methods enableshigher speed, but the price is a degradation in performance.

SUMMARY OF THE INVENTION

The invention enables an effective and precise optimization of rateallocation, without making use of a conventional Lagrangian optimizationscheme.

To implement an optimum rate allocation, the method in accordance withthe invention includes the following steps:

-   -   a) the image (or data) to be compressed is projected onto a base        of localized orthogonal or biorthogonal functions, such as        wavelets,    -   b) a set rate R_(c) is chosen, representing the total number of        bits that can be used to code all coefficients of the        transformed image,    -   c) to allocate this number of bits to the coefficients which are        going to be quantized:        -   c.1 a probability density model in the form of a generalized            Gaussian is associated with each of the subbands,        -   c.2 the parameters of this generalized Gaussian are            estimated while minimizing the relative entropy, or            Kullback-Leibler distance, between this generalized Gaussian            and the empirical distribution of coefficients of each            subband, and an optimum quantization step is then deduced            therefrom, which is such that the total allocated rate R_(c)            is distributed in the various subbands while minimizing the            total distortion.

Generalized Gaussians are described in the article by S. MALLAT: “Theoryfor multiresolution signal decomposition: the wavelet representation”,published in IEEE Transaction on pattern analysis and machineintelligence, vol. 11 (1989) No. 7, pages 674-693.

As a preference, to determine the optimum quantization step for eachsubband, from the parameters of the generalized Gaussian, the graph ofrate as a function of the quantization step and the graph of distortionas a function of the quantization step are determined, and the graphs ofrate and distortion are tabulated and, from these tabulated graphs, theoptimum quantization step is deduced. Minimization of theKullback-Leibler distance for estimating the parameters of thegeneralized Gaussian model ensures a minimization of the cost of codingin accordance with information theory.

Thus, the invention concerns a method for compressing data, inparticular images, in which method this data is projected onto a base oflocalized orthogonal or biorthogonal functions, and which method, toquantize each of the localized functions with a quantization step thatenables an overall set rate R_(c) to be satisfied, includes thefollowing steps:

-   -   a) a probability density model of coefficients in the form of a        generalized Gaussian is associated with each subband,    -   b) the parameters α and β of this density model are estimated        while minimizing the relative entropy, or Kullback-Leibler        distance, between this model and the empirical distribution of        coefficients of each subband, and    -   c) from this model, for each subband, an optimum quantization        step is determined such that the rate allocated is distributed        in the various subbands and such that the total distortion is        minimal.

As a preference, for each subband, the graphs of rate R and distortion Dare deduced, from the parameters α and β, as a function of thequantization step and these graphs are tabulated to determine saidoptimum quantization step.

The transformation is, for example, of the wavelet type.

In this case, according to an embodiment, to determine the parameter βof the generalized Gaussian associated with each subband, the followingexpression is minimized:

${{H\left( p_{1}||G_{{\alpha{(\beta)}},\beta} \right)} = {\frac{1}{\beta} + {\frac{1}{\beta}{\log\left( {\beta\; W_{jk}^{\beta}} \right)}} + {\log\left( \frac{2{\Gamma\left( {1/\beta} \right)}}{\beta} \right)}}},$

in which formula the first term represents said relative entropy,

W_(jk)^(β)has the value:

$W_{jk}^{\beta} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}\;{{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$

W_(jk)[n,m] being a coefficient of a subband, and n_(jk) being thenumber of coefficients in the subband of index j,k.

As a preference, the parameter α of the generalized Gaussian for thecorresponding subband j,k is determined by the following formula:

$\alpha = \sqrt[\beta]{\beta\; W_{jk}^{\beta}}$

The tabulation of the distortion values is advantageously carried outfor a sequence of values of the parameter β.

Similarly, the tabulation of the rates R is advantageously carried outfor a sequence of values of the parameter β.

The sequence of values of the parameters β is for example:

$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}$

As a preference, the bit budget is distributed to each of the subbandsaccording to their ability to reduce the distortion of the compressedimage.

In one embodiment, a bit budget corresponding to the highest tabulatedquantization step is assigned to each subband and the remaining bitbudget is then cut into individual parts that are gradually allocated tothe localized functions having the greatest ability to make the totaldistortion decrease, this operation being repeated until the bit budgetis exhausted.

According to one embodiment, the localized functions and thequantization step for each subband and the parameters of the densitymodel are coded using a lossless entropy coder.

Other features and advantages of the invention will become apparent withthe description of some of its embodiments, this description beingprovided with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1, already described, shows the decomposition into subbands of animage on three resolution levels.

FIG. 2 is a diagram showing graphs of rate for an example application ofthe method according to the invention.

FIG. 3 is a diagram similar to that of FIG. 2 for graphs of distortion.

FIG. 4 is a schematic diagram of a device implementing the methodaccording to the invention to which a control means has been added,which is intended to manage the filling of a buffer memory the role ofwhich is to deliver a constant number of bits to a transmission channelor to any other device requiring a fixed rate.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The example of the invention that will be described below concerns anallocation of rate by subbands. It is carried out in three steps:

First Step: Wavelet Transformation of the Image

This transformation provides a family of wavelet coefficientsW_(jk)[n,m] distributed in various subbands, where j denotes the scaleof the subband and k its orientation (FIG. 1). The number of indexes n,mfor which wavelet coefficients are thus defined depends on the scale ofthe subband. It is therefore noted n_(jk).

Consider, for example, an image of size 512×512 pixels of a terrestriallandscape obtained by an observation satellite. The size of a subbandj,k is in this case n_(jk)=(512/2^(i))².

Second Step: A total rate setting is allocated for all the subbands andthis total rate setting is distributed among the various subbands.

To assign the rate in each subband, the procedure is as follows:

-   -   a) It is considered that the coefficients in each subband j,k        are distributed according to a statistical distribution        corresponding to a generalized Gaussian of parameters α and β.        These parameters α and β are estimated, in a manner that will be        described later, while minimizing the Kullback-Leibler distance        between this statistical model and the empirical distribution of        coefficients of each subband.    -   b) The knowledge of these parameters α and β can be used to        predict, for each subband j,k, the distortion D_(jk) which will        be associated with a rate R_(jk) which will be allocated to it,        and the relationship between these two parameters α and β and        the quantization step Δ_(jk) chosen for this subband. Rate        scheduling is carried out by distributing the rate setting R        among the subbands j,k, in such a way that the complete rate is        allocated to them:

$R = {\sum\limits_{jk}^{\;}R_{jk}}$

and that the resulting distortion

$D = {\sum\limits_{jk}^{\;}D_{jk}}$is minimal.

At the end of the scheduling phase, an optimum quantization step Δ_(jk)has therefore been determined for each subband, enabling the ratesetting R to be reached while minimizing the total distortion D.

Third step: The subbands are quantized with quantization steps Δ_(jk)which can be different from one subband to another and the quantizedcoefficients are sent to an entropy coder.

Determination of Parameters α and β of the Generalized Gaussians foreach Subband:

For each subband j,k, the statistical distribution of the subband tendstoward a generalized Gaussian G_(αβ):

${G_{\alpha\beta}(x)} = {A_{\alpha\beta}{\mathbb{e}}^{- {\frac{x}{\alpha}}^{\beta}}}$where

$A_{\alpha\beta} = \frac{\beta}{2{{\alpha\Gamma}\left( {1/\beta} \right)}}$

To determine the parameters α and β, as indicated above, the relativeentropy between this subband density model and the empirical density ofthis subband is minimized.

It is recalled here that the relative entropy, or Kullback-Leiblerdistance, between two probability densities p₁ and p₂ is given by:

${D\left( p_{1}||p_{2} \right)} = {\int{{p_{1}(x)}\log\frac{p_{1}(x)}{p_{2}(x)}{{\mathbb{d}x}.}}}$

In the sense of this Kullback-Leibler distance, the distribution p₂which best tends toward the distribution p₁ is that which minimizesD(p₁∥p₂).

Note furthermore that to determine p₂ which minimizes D(p₁∥p₂), for afixed p₁, the following is minimized:D(p ₁ ∥p ₂)=∫p ₁(x)log p ₁(x)dx−∫p ₁(x)log p ₂(x)dx.

As the first term of this difference does not depend on p₂, minimizingthis sum in p₂ amounts to minimizing the second term of this sum, andtherefore amounts to minimizing:H(p ₁ ∥p ₂)=−∫p ₁(x)log p ₂(x)dx  (1)

If p₁ and p₂ are discrete distributions, the term above is then theaverage rate of coding of a source of symbols of probabilitydistribution p₁, coded with optimum entropy symbols for a distributionp₂.

Minimizing this term amounts therefore to choosing a model distributionp₂ which will produce the most efficient symbols for coding adistribution source p₁.

Therefore D(p₁∥p₂) will be minimized and not the reverse, D(p₂∥p₁),since the Kullback-Leibler distance is not symmetric.

In the present case, the distribution p₁ is the empirical distributionof a subband (j,k):

$\begin{matrix}{{p_{1}(x)} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}\;{\delta\left( {x - {W_{jk}\left\lbrack {n,m} \right\rbrack}} \right)}}}} & (2)\end{matrix}$

and the distribution p₂ is the generalized Gaussian indicated above:

$\begin{matrix}{{p_{2}(x)} = {{G_{\alpha\beta}(x)} = {A_{\alpha\beta}{\mathbb{e}}^{- {\frac{x}{\alpha}}^{\beta}}}}} & (3)\end{matrix}$

The expression (1) calculated with (2) and (3) gives:

$\begin{matrix}{{H\left( p_{1}||G_{\alpha\beta} \right)} = {{- \frac{1}{n_{jk}}}{\sum\limits_{n,m}^{\;}{\log\left( {A_{\alpha\beta}{\mathbb{e}}^{- {\frac{W_{jk}{\lbrack{n,m}\rbrack}}{\alpha}}^{\beta}}} \right)}}}} \\{= {{{- \log}\; A_{\alpha\beta}} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}}}\end{matrix}$

Using the value of A_(αβ) gives:

$\begin{matrix}{{H\left( p_{1}||G_{\alpha\beta} \right)} = {{{- \log}\;\beta} + {\log\; 2} + {\log\;\alpha} + {\log\;{\Gamma\left( {1/\beta} \right)}} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}\;{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}}} & (4)\end{matrix}$

Minimization in α,β is carried out in two steps: during the first step,α is minimized for a fixed β; this minimization is performed by a simpleexplicit calculation. During the second step, a search is carried outfor an optimum β using tabulated values to avoid calculations that aretoo complex.

Minimization of α for a Fixed β

To minimize expression (4) in α, with a fixed β, the sum of the termsthat depend on α must be minimized, other terms being constants, thatis:

${\log\;\alpha} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}$

The derivative of this expression with respect to α is:

$\frac{1}{\alpha} - {\frac{\beta}{\alpha^{\beta + 1}n_{jk}}{\sum\limits_{n,m}^{\;}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}$and its derivative is cancelled for

$\alpha = \sqrt[\beta]{\frac{\beta\;}{n_{jk}}{\sum\limits_{n,m}^{\;}{{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$

To lighten the notations, the moment of order β of the subband j,k isdefined:

$W_{jk}^{\beta} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}{{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$which is therefore the average of the absolute values of coefficients ofthe subband, raised to the power β.

The optimum α is then written simply:

$\begin{matrix}{\alpha = \sqrt[\beta]{\beta\; W_{jk}^{\beta}}} & (5)\end{matrix}$

Calculation of Optimum β

It is therefore known how to determine the optimum α once β is known. Todetermine the optimum β, in equation (4) α is replaced by the valuegiven by equation (5). The following is obtained:

$\begin{matrix}{{H\left( p_{1}||G_{a\beta} \right)} = {{{- \log}\;\beta} + {\log\; 2} + {\frac{1}{\beta}{\log\left( {\beta\; W_{jk}^{\beta}} \right)}} + {\log\;{\Gamma\left( {1/\beta} \right)}} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}\;{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\sqrt[\beta]{\beta\; W_{jk}^{\beta}}}}^{\beta}}}}} \\{= {{{- \log}\;\beta} + {\log\; 2} + {\frac{1}{\beta}{\log\left( {\beta\; W_{jk}^{\beta}} \right)}} + {\log\;{\Gamma\left( {1/\beta} \right)}} + \frac{W_{jk}^{\beta}}{\beta\; W_{jk}^{\beta}}}} \\{= {{{- \log}\;\beta} + {\log\; 2} + {\frac{1}{\beta}{\log\left( {\beta\; W_{jk}^{\beta}} \right)}} + {\log\;{\Gamma\left( {1/\beta} \right)}} + \frac{1}{\beta}}}\end{matrix}$

The optimum value of β will therefore be obtained by minimizing theexpression above which can again be rewritten in the following form:

$\begin{matrix}{{H\left( {p_{1}{G_{{a{(\beta)}},\beta}}} \right)} = {\frac{1}{\beta} + {\frac{1}{\beta}{\log\left( {\beta\; W_{jk}^{\beta}} \right)}} + {\log\left( \frac{2{\Gamma\left( {1/\beta} \right)}}{\beta} \right)}}} & (6)\end{matrix}$

Thus, the calculation of the optimum α,β pair is carried out in twostages:

First, β is calculated by minimizing expression (6).

Secondly, α is calculated using expression (5).

β is chosen from a finite number of candidate values, for example

$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}$and most of the calculations can be tabulated. The only component whichactually depends on the subband is

$\frac{1}{\beta}{{\log\left( {\beta\; W_{jk}^{\beta}} \right)}.}$

The calculation of α is hence explicit.

To avoid any confusion, it is stated here that, strictly, thesecoefficients should be called α_(jk) and β_(jk) since they are, apriori, different for each subband.

In the example above, for the subband W_(1,1) the calculation of W_(1,1)^(β) and of the expression (6) for values of β in the set

$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}$gives the values set out in table 1 below:

TABLE 1Values  of  W_(1, 1)^(β)  and  H(p₁||G_(α(β), β))  for  the  candidate  values  of  β.β 1/2 $1/\sqrt{2}$ 1 $\sqrt{2}$ 2 W_(1, 1)^(β) 1.42 1.88 3.17 7.96 38.6H(p₁||G_(α(β), β)) 2.70 2.74 2.85 3.02 3.25

The minimum is therefore reached for β=½. The corresponding value of αis then

$\alpha = {\left( \frac{W_{1,1}^{1/2}}{2} \right)^{2} = 0.502}$

The following is therefore obtained: α_(1,1)=0.502 and β_(1,1)=½.

The parameters α and β for the other subbands are calculated in the sameway.

After having determined α and β, the relationship between rate R andquantization step Δ and between the distortion D and the quantizationstep are determined. The graphs R(Δ) and D(Δ) are tabulated and used insuch a way that, for each subband, an optimum quantization step isobtained, that is, in such a way that the allocated rate R isdistributed in the various subbands and that the total distortion isminimal.

Prediction of the Relationship between Rate and Quantization Step

If a subband has a statistical distribution described by a generalizedGaussian G_(α,β), the associated rate (in bits per coefficient) is notedr(α,β,Δ), for a quantization step Δ.

This rate is the entropy of the quantized subband with a quantizationstep Δ, which is, for example in the case of a uniform quantization:

${r\left( {\alpha,\beta,\Delta} \right)} = {- {\sum\limits_{k = {- \infty}}^{+ \infty}\;{{p_{k}\left( {\alpha,\beta,\Delta} \right)}\log_{2}{p_{k}\left( {\alpha,\beta,\Delta} \right)}}}}$

if

p_(k)(α, β, Δ) = ∫_((k − 1/2)Δ)^((k + 1/2)Δ)G_(α, β)(x) 𝕕x

The rate allocation means must know the relationship between r and(α,β,Δ). For this purpose, it is sufficient to tabulate the function r.Since it can be verified that:

${r\left( {\alpha,\beta,\Delta} \right)} = {r\left( {1,\beta,\frac{\Delta}{\alpha}} \right)}$it is sufficient to tabulate the functions x

r(1,β,x) for all the candidate values of β (there are five in the aboveexample). The x values are also considered in a range [x_(min),x_(max)].

Note that the same type of calculations and tabulations can be performedif a quantizer is used, having a quantization interval, centered at 0,of different size, as is often the case in the coding of images bywavelets.

For a subband of size n_(jk), the total rate will then beR _(jk) =n _(jk) r(α,β,Δ).

FIG. 2 indicates tabulated values of r(1,β,x), for values of β in

$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}.$

In FIG. 2, the values of x are plotted as abscissae and the values r ofrate are plotted as ordinates.

Relationship between Distortion and Quantization Step

In the same way, the relationship between quantization step anddistortion can be modeled in a subband having a statistical distributiontending toward a generalized Gaussian. The average distortion percoefficient is noted D(α,β,Δ), for a subband of generalized Gaussianstatistical distribution G_(αβ).

This distortion is written:

${\mathbb{d}\left( {\alpha,\beta,\Delta} \right)} = {\int_{- \infty}^{+ \infty}{{G_{\alpha\beta}(x)}\left( {x - {\Delta\left\lbrack \frac{x}{\Delta} \right\rbrack}} \right)^{2}\ {\mathbb{d}x}}}$

where [x] denotes the integer that is closest to x, for the case inwhich the quantization operator is a uniform quantization.

Here also, the values of d(α,β,Δ) can be tabulated economically, bymaking use of the following homogeneity equation:d(α,β,Δ)=α² d(1,β,Δ/α)

and it will therefore be sufficient to tabulate the functionx

d(1,β,x)

for values of β in the range of chosen candidate values, and values of xselected in [x_(min),x_(max)].

Here again, the total distortion for a subband is the sum of distortionsper coefficients, and it is therefore written:D _(jk) =n _(jk) d(α,β,Δ)

The tabulated graphs are shown in FIG. 3.

In said FIG. 3, the x values are plotted as abscissa and the distortionas ordinate.

For a fixed α and β, the relationship between r and Δ is invertible.This inversion is performed either by tabulation in advance, or byinterpolation of the tabulated values of r(1,β,x). Similarly, therelationship between d and Δ can be inverted.

The function which associates a quantization step Δ with a distortion dis noted d⁻¹ and the function which associates the quantization step Δwith a rate r is noted r⁻¹, for fixed α and β:Δ=d ⁻¹(α,β,d)Δ=r ⁻¹(α,β,r)

The rate scheduling consists in cutting the set rate R into fragmentswhich will be promptly allocated. This assignment is carried outiteratively.

For the initialization, the starting point, for each subband, is amaximum quantization step specified by the tables of R(α,β,Δ) andD(α,β,Δ), which will be:

$\Delta_{jk}{\frac{x_{\max}}{\alpha_{jk}}.}$

Rates R_(jk) and distortions D_(jk) are associated with thesequantization steps by the formulae:R _(jk) =n _(jk) r(α_(jk),β_(jk),Δ_(jk))D _(jk) =n _(jk) d(α_(jk),β_(jk),Δ_(jk))

The rate remaining to be allocated is therefore

$R - {\sum\limits_{jk}^{\;}\;{R_{jk}.}}$This rate is cut into N fragments F_(n) which can be of the same size orof different sizes, with

${F_{1} + \mspace{11mu}\ldots\mspace{11mu} + F_{N}} = {R - {\sum\limits_{jk}^{\;}\;{R_{jk}.}}}$

Above all, the maximum size of a fragment must remain small in view ofthe total rate setting. Each of these fragments is then allocatediteratively, as follows:

For each subband, the potential rate is calculated which would be therate associated with the subband if this new rate fragment F_(n)happened to be allocated to it:

R_(jk)^(*) = R_(jk) + F_(n).

The associated potential quantization steps are then calculated:

$\Delta_{jk}^{*} = {r^{- 1}\left( {\alpha_{jk},\beta_{jk},\frac{R_{jk}^{*}}{n_{jk}}} \right)}$

then the new associated distortions:

D_(jk)^(*) = n_(jk)d(α_(jk), β_(jk), Δ_(jk)^(*))

For each subband j,k it can be estimated what the reduction indistortion would be if the rate fragment F were to be allocated to it.This reduction would be:

D_(jk) − D_(jk)^(*)

The “best” subband j,k, that is the one for which the reduction indistortion is strongest, is allocated the fragment:

R_(jk) ← R_(jk)^(*) D_(jk) ← D_(jk)^(*) Δ_(jk) ← Δ_(jk)^(*)

The same rate allocation is reiterated for the next fragments F_(n+1),F_(n+2), until the rate to be allocated is exhausted.

Table 2 below indicates, for the abovementioned example, which are thevalues of α and β retained. The coding of the low-pass subband W_(3,0)is performed in DPCM (Difference Pulse Code Modulation), and thestatistics indicated in the table are therefore statistics ofW_(3,0)[n_(k),m_(k)]−W_(3,0)[n_(k-1),m_(k-1)], where the numbering ofpairsk

(n_(k),m_(k))indicates the direction chosen for the low-pass coefficients during thecoding.

TABLE 2 Laplacian parameters for a test image j k α_(jk) β_(jk) D_(jk)R_(jk) 1 1 0.50 1/2 50 444 1 2 0.35 1/2 35 444 1 3 0.56 1√{square rootover (2)} 56 0 2 1 1.50 1/2 150 111 2 2 1.20 1/2 120 111 2 3 0.75 1/2 75111 3 1 3.60 1/2 360 27.8 3 2 3.40 1/2 340 27.8 3 3 2.10 1/2 210 27.8 30 11 1/2 1100 27.8

The total rate allocated is therefore at least the sum of the ratesR_(jk) above, that is 1332 bits, therefore 0.005 bits per pixel. Thesetting is of 1 bit per pixel, that is a total budget of R=262144 bits,the image being of size 512×512.

The remaining rate to be allocated of 260812 bits is divided into partsthat are not necessarily equal, but of sizes that are all less than afixed limit. The first rate fragment to be allocated is for example of384 bits.

The tables are used to calculate, for each subband, what the distortionassociated with each subband will be if a budget of 384 additional bitswere to be attributed to it. The result of these calculations is set outin table 3.

For example, for the subband W_(1,2), the total rate R_(1,2) wouldchange from 444 bits to 828 bits, and the distortion D_(1,2) wouldchange from 0.93×10⁶ to 0.89×10⁶ and would therefore be reduced by0.03×10⁶.

TABLE 3 Simulations for allocating the first fragment. The distortionsare to be multiplied by 10⁶. j k R_(jk) Δ_(jk) D_(jk) R*_(jk) Δ*_(jk)D*_(jk) D_(jk) − D*_(jk) 1 1 444 50 1.86 828 43 1.79 0.07 1 2 444 350.93 828 30 0.89 0.04 1 3 0 35 0.19 384 12 0.18 0.01 2 1 111 150 4.01495 98 3.56 0.45 2 2 111 120 2.71 495 80 2.40 0.31 2 3 111 75 1.03 49549 0.91 0.12 3 1 27.8 360 5.85 411 154 4.11 1.74 3 2 27.8 340 5.38 411148 3.78 1.60 3 3 27.8 210 1.96 411 89 1.37 0.59 3 0 27.8 1100 5.24 411461 3.68 1.56

In table 3, it is seen that the strongest reduction is obtained byallocating the 384 bits to the subband W_(3,1). Therefor Δ_(3,1),R_(3,1) and D_(3,1) are updated and the process is continued by theallocation of the next fragment. When all the fragments have beenallocated, quantization setting Δ_(jk), and prediction on the associatedrate R_(jk) and distortion D_(j) for each subband, are thereforeobtained.

The table finally obtained for the image is given in table 4 below:

TABLE 4 Example of settings of quantization and associated rates by thescheduler. The rates are given in bits per coefficient. j k Δ_(jk)R_(jk) 1 1 6.1 0.92 1 2 6.5 0.53 1 3 8.4 0.03 2 1 5.7 2.5 2 2 5.7 2.1 23 5.8 1.4 3 0 6.3 3.8 3 1 5.8 3.7 3 2 6 2.9 3 3 5.9 5.5

The allocation of bits can be performed in open-loop mode or,preferably, in closed-loop mode with a device of the type illustrated inFIG. 4.

The rate regulation obtained using the device illustrated in FIG. 4 isto the nearest bit. This device is based on the one described in theFrench patent filed on Mar. 18, 1999 under the number 99/03371.

In this patent, a technique is described for the acquisition of imagesof the Earth by a moving observation satellite (“Push Broom” mode) inwhich the images are compressed and transmitted to the ground via atransmission channel which imposes a constant rate. To achieve thisobjective, an optimum allocation of bits is performed, in real time orin slightly deferred time, in order to code the subband coefficients,and then the allocation errors and variations in rate at the output ofthe entropy coder are corrected.

This device includes a compression means 20, and the image data isapplied to the input 22 of said compression means 20. The compressionmeans 20 includes a wavelet transformation unit 24 supplying data to asubband quantizer which delivers data to a coder 28 the output of whichforms the output of the compression means 20. This output of thecompression means 20 is linked to a regulation buffer memory 30supplying, at its output, compressed data according to a rate R_(c).

The coder 28 also supplies a number N_(p) of bits produced which isapplied to an input 33 of a control unit 32 having another input for theset rate R_(c) and an output applied to a first input of a bitallocation unit 34. This latter unit 34 has another input 36 to whichthe set rate R_(c) is applied and an input 38 to which the output datafrom the transformation unit 24 is applied.

The output of the unit 34 is applied to an input 40 of the subbandquantizer 26.

The method and device in accordance with the invention enable the costof coding the image to be reduced and ensure a rapid rate regulationsince, unlike in the prior art, no iterative optimization (Lagrangianfor example) is performed. The calculations are simple. Thus, the methodin accordance with the invention is well suited for all the applicationsand, in particular, for space applications in which the resources arenecessarily limited.

It is to be noted that, in the present description, the term “image”must be understood as being in the sense of an image of at least onedimension, that is to say that the invention extends to data in general.

1. A computer implemented method for compressing data by transform, inwhich method the data is projected onto a base of localized orthogonalor biorthogonal functions, and each of the localized functions isquantize to enable an overall set rate R_(c) to be satisfied, the methodcomprising: associating a respective probability density model ofcoefficients in the form of a generalized Gaussian with each of aplurality of subbands, estimating parameter β of the density model,while minimizing a relative entropy, or Kullback-Leibler distance,between the model and an empirical distribution of coefficients of eachof the plurality of subbands, estimating parameter α based on theestimating parameter β, and determining and allocating a respective rateto each of the plurality of subbands such that a total distortion isminimal.
 2. The computer implemented method claimed in claim 1, whereinsaid localized orthogonal or biorthogonal functions are wavelets.
 3. Thecomputer implemented method claimed in claim 2, wherein, to determinesaid respective parameter β of the generalized Gaussian corresponding toa subband of index j,k, the following expression is minimized:${H\left( p_{1}||G_{{\alpha{(\beta)}},\beta} \right)} = {\frac{1}{\beta} + {\frac{1}{\beta}{\log\left( {\beta\; W_{jk}^{\beta}} \right)}} + {\log\left( \frac{2{\Gamma\left( {1/\beta} \right)}}{\beta} \right)}}$in which formula the first term represents said relative entropy,W_(jk)^(β)  has the value:$W_{jk}^{\beta} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\;}{{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$W_(jk) [n,m] being a coefficient of the subband of index j,k, and n_(jk)being the number of coefficients in the subband of index j,k.
 4. Thecomputer implemented method claimed in claim 3, wherein said parameter αof the generalized Gaussian corresponding to the subband of index j,k isdetermined by the following formula:$\alpha = {\sqrt[\beta]{\beta\; W_{jk}^{\beta}}.}$
 5. The computerimplemented method claimed in claim 1, wherein, for each of theplurality of subbands, graphs of rate R and distortion D are deduced,from the respective parameters α and β, as a function of thequantization and said graphs are tabulated to determine the respectiverate.
 6. The computer implemented method claimed in claim 5, wherein thetabulation of the distortion D is carried out for a sequence of valuesof the parameter β.
 7. The computer implemented method claimed in claim5, wherein the tabulation of the rate R is carried out for a sequence ofvalues of the parameter β.
 8. The computer implemented method claimed inclaim 6, wherein the sequence of values of the parameters β is$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,\sqrt{2},2} \right\}.$
 9. Thecomputer implemented method claimed in claim 1, wherein a total bitbudget is distributed to each of the plurality of subbands according torespective ability of each of the plurality of subbands to reduce thedistortion of the compressed image.
 10. The computer implemented methodclaimed in claim 9, wherein a bit budget corresponding to the highesttabulated quantization is assigned to each of the plurality of subbandsand wherein remain of the bit budget is then cut into individual partsthat are gradually allocated to the localized functions having thegreatest ability to make the total distortion decrease, this operationbeing repeated until the bit budget is exhausted.
 11. The computerimplemented method claimed in claim 1, wherein the localized functionsand the quantization for each of the plurality of subbands and theparameters α and β of the density model are coded using a losslessentropy coder.